fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
inv_mem_connectedComponent_one {G : Type*} [TopologicalSpace G] [DivisionMonoid G]
[ContinuousInv G] {g : G} (hg : g ∈ connectedComponent (1 : G)) :
g⁻¹ ∈ connectedComponent (1 : G) := by
rw [← inv_one]
exact
Continuous.image_connectedComponent_subset continuous_inv _
((Set.mem_image _ _ _).mp ⟨g,... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | inv_mem_connectedComponent_one | null |
@[to_additive /-- The connected component of 0 is a subgroup of `G`. -/]
Subgroup.connectedComponentOfOne (G : Type*) [TopologicalSpace G] [Group G]
[IsTopologicalGroup G] : Subgroup G where
carrier := connectedComponent (1 : G)
one_mem' := mem_connectedComponent
mul_mem' hg hh := mul_mem_connectedComponent_o... | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.connectedComponentOfOne | The connected component of 1 is a subgroup of `G`. |
@[to_additive
/-- An additive monoid homomorphism (a bundled morphism of a type that implements
`AddMonoidHomClass`) from an additive topological group to an additive topological monoid is
continuous provided that it is continuous at zero. See also
`uniformContinuous_of_continuousAt_zero`. -/]
continuous_of_con... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_of_continuousAt_one | If a subgroup of a topological group is commutative, then so is its topological closure.
See note [reducible non-instances]. -/
@[to_additive
/-- If a subgroup of an additive topological group is commutative, then so is its
topological closure.
See note [reducible non-instances]. -/]
abbrev Subgroup.commGroupTopolo... |
continuous_of_continuousAt_one₂ {H M : Type*} [CommMonoid M] [TopologicalSpace M]
[ContinuousMul M] [Group H] [TopologicalSpace H] [IsTopologicalGroup H] (f : G →* H →* M)
(hf : ContinuousAt (fun x : G × H ↦ f x.1 x.2) (1, 1))
(hl : ∀ x, ContinuousAt (f x) 1) (hr : ∀ y, ContinuousAt (f · y) 1) :
Continu... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_of_continuousAt_one₂ | null |
IsTopologicalGroup.isInducing_iff_nhds_one
{H : Type*} [Group H] [TopologicalSpace H] [IsTopologicalGroup H] {F : Type*}
[FunLike F G H] [MonoidHomClass F G H] {f : F} :
Topology.IsInducing f ↔ 𝓝 (1 : G) = (𝓝 (1 : H)).comap f := by
rw [Topology.isInducing_iff_nhds]
refine ⟨(map_one f ▸ · 1), fun hf x ... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.isInducing_iff_nhds_one | null |
IsTopologicalGroup.isOpenMap_iff_nhds_one
{H : Type*} [Monoid H] [TopologicalSpace H] [ContinuousConstSMul H H]
{F : Type*} [FunLike F G H] [MonoidHomClass F G H] {f : F} :
IsOpenMap f ↔ 𝓝 1 ≤ .map f (𝓝 1) := by
refine ⟨fun H ↦ map_one f ▸ H.nhds_le 1, fun h ↦ IsOpenMap.of_nhds_le fun x ↦ ?_⟩
have : F... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.isOpenMap_iff_nhds_one | null |
@[to_additive /-- Let `A` and `B` be topological additive groups, and let `φ : A → B` be a
continuous surjective additive group homomorphism. Assume furthermore that `φ` is a quotient map
(i.e., `V ⊆ B` is open iff `φ⁻¹ V` is open). Then `φ` is an open quotient map, and in particular an
open map. -/]
MonoidHom.isOpenQu... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | MonoidHom.isOpenQuotientMap_of_isQuotientMap | Let `A` and `B` be topological groups, and let `φ : A → B` be a continuous surjective group
homomorphism. Assume furthermore that `φ` is a quotient map (i.e., `V ⊆ B`
is open iff `φ⁻¹ V` is open). Then `φ` is an open quotient map, and in particular an open map. |
IsTopologicalGroup.ext {G : Type*} [Group G] {t t' : TopologicalSpace G}
(tg : @IsTopologicalGroup G t _) (tg' : @IsTopologicalGroup G t' _)
(h : @nhds G t 1 = @nhds G t' 1) : t = t' :=
TopologicalSpace.ext_nhds fun x ↦ by
rw [← @nhds_translation_mul_inv G t _ _ x, ← @nhds_translation_mul_inv G t' _ _ x, ... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.ext | null |
IsTopologicalGroup.ext_iff {G : Type*} [Group G] {t t' : TopologicalSpace G}
(tg : @IsTopologicalGroup G t _) (tg' : @IsTopologicalGroup G t' _) :
t = t' ↔ @nhds G t 1 = @nhds G t' 1 :=
⟨fun h => h ▸ rfl, tg.ext tg'⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.ext_iff | null |
ContinuousInv.of_nhds_one {G : Type*} [Group G] [TopologicalSpace G]
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (fun x : G => x₀ * x) (𝓝 1))
(hconj : ∀ x₀ : G, Tendsto (fun x : G => x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)) : ContinuousInv G := by
refine ⟨continuous_iff_continuo... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | ContinuousInv.of_nhds_one | null |
IsTopologicalGroup.of_nhds_one' {G : Type u} [Group G] [TopologicalSpace G]
(hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1))
(hright : ∀ x₀ : G, 𝓝 x₀ = map (fun x => x * x₀) ... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.of_nhds_one' | null |
IsTopologicalGroup.of_nhds_one {G : Type u} [Group G] [TopologicalSpace G]
(hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (x₀ * ·) (𝓝 1))
(hconj : ∀ x₀ : G, Tendsto (x₀ * · * x₀⁻¹) (𝓝 1) (𝓝 1)) : ... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.of_nhds_one | null |
IsTopologicalGroup.of_comm_of_nhds_one {G : Type u} [CommGroup G] [TopologicalSpace G]
(hmul : Tendsto (uncurry ((· * ·) : G → G → G)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hinv : Tendsto (fun x : G => x⁻¹) (𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : G, 𝓝 x₀ = map (x₀ * ·) (𝓝 1)) : IsTopologicalGroup G :=
IsTopologicalGroup.of_nh... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.of_comm_of_nhds_one | null |
@[to_additive
/-- Any first countable topological additive group has an antitone neighborhood basis
`u : ℕ → set G` for which `u (n + 1) + u (n + 1) ⊆ u n`.
The existence of such a neighborhood basis is a key tool for `QuotientAddGroup.completeSpace` -/]
IsTopologicalGroup.exists_antitone_basis_nhds_one [FirstCou... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.exists_antitone_basis_nhds_one | Any first countable topological group has an antitone neighborhood basis `u : ℕ → Set G` for
which `(u (n + 1)) ^ 2 ⊆ u n`. The existence of such a neighborhood basis is a key tool for
`QuotientGroup.completeSpace` |
@[to_additive const_sub]
Filter.Tendsto.const_div' (b : G) {c : G} {f : α → G} {l : Filter α}
(h : Tendsto f l (𝓝 c)) : Tendsto (fun k : α => b / f k) l (𝓝 (b / c)) :=
tendsto_const_nhds.div' h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.Tendsto.const_div' | null |
Filter.tendsto_const_div_iff {G : Type*} [CommGroup G] [TopologicalSpace G] [ContinuousDiv G]
(b : G) {c : G} {f : α → G} {l : Filter α} :
Tendsto (fun k : α ↦ b / f k) l (𝓝 (b / c)) ↔ Tendsto f l (𝓝 c) := by
refine ⟨fun h ↦ ?_, Filter.Tendsto.const_div' b⟩
convert h.const_div' b with k <;> rw [div_div_ca... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.tendsto_const_div_iff | null |
Filter.Tendsto.div_const' {c : G} {f : α → G} {l : Filter α} (h : Tendsto f l (𝓝 c))
(b : G) : Tendsto (f · / b) l (𝓝 (c / b)) :=
h.div' tendsto_const_nhds | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.Tendsto.div_const' | null |
Filter.tendsto_div_const_iff {G : Type*}
[CommGroupWithZero G] [TopologicalSpace G] [ContinuousDiv G]
{b : G} (hb : b ≠ 0) {c : G} {f : α → G} {l : Filter α} :
Tendsto (f · / b) l (𝓝 (c / b)) ↔ Tendsto f l (𝓝 c) := by
refine ⟨fun h ↦ ?_, fun h ↦ Filter.Tendsto.div_const' h b⟩
convert h.div_const' b⁻¹ ... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.tendsto_div_const_iff | null |
Filter.tendsto_sub_const_iff {G : Type*}
[AddCommGroup G] [TopologicalSpace G] [ContinuousSub G]
(b : G) {c : G} {f : α → G} {l : Filter α} :
Tendsto (f · - b) l (𝓝 (c - b)) ↔ Tendsto f l (𝓝 c) := by
refine ⟨fun h ↦ ?_, fun h ↦ Filter.Tendsto.sub_const h b⟩
convert h.sub_const (-b) with k <;> rw [sub_... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Filter.tendsto_sub_const_iff | null |
continuous_div_left' (a : G) : Continuous (a / ·) := continuous_const.div' continuous_id
@[to_additive (attr := continuity) continuous_sub_right] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_div_left' | null |
continuous_div_right' (a : G) : Continuous (· / a) := continuous_id.div' continuous_const | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | continuous_div_right' | null |
@[to_additive (attr := simps! +simpRhs)
/-- A version of `Homeomorph.addLeft a (-b)` that is defeq to `a - b`. -/]
Homeomorph.divLeft (x : G) : G ≃ₜ G :=
{ Equiv.divLeft x with
continuous_toFun := continuous_const.div' continuous_id
continuous_invFun := continuous_inv.mul continuous_const }
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.divLeft | A version of `Homeomorph.mulLeft a b⁻¹` that is defeq to `a / b`. |
isOpenMap_div_left (a : G) : IsOpenMap (a / ·) :=
(Homeomorph.divLeft _).isOpenMap
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_div_left | null |
isClosedMap_div_left (a : G) : IsClosedMap (a / ·) :=
(Homeomorph.divLeft _).isClosedMap | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedMap_div_left | null |
@[to_additive (attr := simps! +simpRhs)
/-- A version of `Homeomorph.addRight (-a) b` that is defeq to `b - a`. -/]
Homeomorph.divRight (x : G) : G ≃ₜ G :=
{ Equiv.divRight x with
continuous_toFun := continuous_id.div' continuous_const
continuous_invFun := continuous_id.mul continuous_const }
@[to_additive] | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.divRight | A version of `Homeomorph.mulRight a⁻¹ b` that is defeq to `b / a`. |
isOpenMap_div_right (a : G) : IsOpenMap (· / a) := (Homeomorph.divRight a).isOpenMap
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isOpenMap_div_right | null |
isClosedMap_div_right (a : G) : IsClosedMap (· / a) := (Homeomorph.divRight a).isClosedMap
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedMap_div_right | null |
tendsto_div_nhds_one_iff {α : Type*} {l : Filter α} {x : G} {u : α → G} :
Tendsto (u · / x) l (𝓝 1) ↔ Tendsto u l (𝓝 x) :=
haveI A : Tendsto (fun _ : α => x) l (𝓝 x) := tendsto_const_nhds
⟨fun h => by simpa using h.mul A, fun h => by simpa using h.div' A⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | tendsto_div_nhds_one_iff | null |
nhds_translation_div (x : G) : comap (· / x) (𝓝 1) = 𝓝 x := by
simpa only [div_eq_mul_inv] using nhds_translation_mul_inv x | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | nhds_translation_div | null |
@[to_additive]
IsTopologicalGroup.t1Space (h : @IsClosed G _ {1}) : T1Space G :=
⟨fun x => by simpa using isClosedMap_mul_right x _ h⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | IsTopologicalGroup.t1Space | null |
@[to_additive
/-- A subgroup `S` of an additive topological group `G` acts on `G` properly
discontinuously on the left, if it is discrete in the sense that `S ∩ K` is finite for all compact
`K`. (See also `DiscreteTopology`. -/]
Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite (S : Subgroup G)
(hS : Ten... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.properlyDiscontinuousSMul_of_tendsto_cofinite | A subgroup `S` of a topological group `G` acts on `G` properly discontinuously on the left, if
it is discrete in the sense that `S ∩ K` is finite for all compact `K`. (See also
`DiscreteTopology`.) |
@[to_additive
/-- A subgroup `S` of an additive topological group `G` acts on `G` properly discontinuously
on the right, if it is discrete in the sense that `S ∩ K` is finite for all compact `K`.
(See also `DiscreteTopology`.)
If `G` is Hausdorff, this can be combined with `t2Space_of_properlyDiscontinuousVAdd_... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Subgroup.properlyDiscontinuousSMul_opposite_of_tendsto_cofinite | A subgroup `S` of a topological group `G` acts on `G` properly discontinuously on the right, if
it is discrete in the sense that `S ∩ K` is finite for all compact `K`. (See also
`DiscreteTopology`.)
If `G` is Hausdorff, this can be combined with `t2Space_of_properlyDiscontinuousSMul_of_t2Space`
to show that the quotie... |
@[to_additive
/-- Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
`0` such that `K + V ⊆ U`. -/]
compact_open_separated_mul_right {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
(hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), K * V ⊆ U := by
refine hK.induction_on ?_ ?_ ?_ ?_
· exact... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | compact_open_separated_mul_right | Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of `1`
such that `K * V ⊆ U`. |
@[to_additive
/-- Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of
`0` such that `V + K ⊆ U`. -/]
compact_open_separated_mul_left {K U : Set G} (hK : IsCompact K) (hU : IsOpen U)
(hKU : K ⊆ U) : ∃ V ∈ 𝓝 (1 : G), V * K ⊆ U := by
rcases compact_open_separated_mul_right (hK.i... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | compact_open_separated_mul_left | Given a compact set `K` inside an open set `U`, there is an open neighborhood `V` of `1`
such that `V * K ⊆ U`. |
@[to_additive
/-- A compact set is covered by finitely many left additive translates of a set
with non-empty interior. -/]
compact_covered_by_mul_left_translates {K V : Set G} (hK : IsCompact K)
(hV : (interior V).Nonempty) : ∃ t : Finset G, K ⊆ ⋃ g ∈ t, (g * ·) ⁻¹' V := by
obtain ⟨t, ht⟩ : ∃ t : Finset G, ... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | compact_covered_by_mul_left_translates | A compact set is covered by finitely many left multiplicative translates of a set
with non-empty interior. |
@[to_additive
/-- Given two compact sets in a noncompact additive topological group, there is a
translate of the second one that is disjoint from the first one. -/]
exists_disjoint_smul_of_isCompact [NoncompactSpace G] {K L : Set G} (hK : IsCompact K)
(hL : IsCompact L) : ∃ g : G, Disjoint K (g • L) := by
hav... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | exists_disjoint_smul_of_isCompact | Every weakly locally compact separable topological group is σ-compact.
Note: this is not true if we drop the topological group hypothesis. -/
@[to_additive SeparableWeaklyLocallyCompactAddGroup.sigmaCompactSpace
/-- Every weakly locally compact separable topological additive group is σ-compact.
Note: this is not ... |
@[to_additive]
nhds_mul (x y : G) : 𝓝 (x * y) = 𝓝 x * 𝓝 y :=
calc
𝓝 (x * y) = map (x * ·) (map (· * y) (𝓝 1 * 𝓝 1)) := by simp
_ = map₂ (fun a b => x * (a * b * y)) (𝓝 1) (𝓝 1) := by rw [← map₂_mul, map_map₂, map_map₂]
_ = map₂ (fun a b => x * a * (b * y)) (𝓝 1) (𝓝 1) := by simp only [mul_assoc]... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | nhds_mul | null |
@[to_additive (attr := simps)
/-- On an additive topological group, `𝓝 : G → Filter G` can be promoted to an `AddHom`. -/]
nhdsMulHom : G →ₙ* Filter G where
toFun := 𝓝
map_mul' _ _ := nhds_mul _ _ | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | nhdsMulHom | On a topological group, `𝓝 : G → Filter G` can be promoted to a `MulHom`. |
@[to_additive /-- If `G` is an additive group with topological negation, then it is homeomorphic to
its additive units. -/]
toUnits_homeomorph [Group G] [TopologicalSpace G] [ContinuousInv G] : G ≃ₜ Gˣ where
toEquiv := toUnits.toEquiv
continuous_toFun := Units.continuous_iff.2 ⟨continuous_id, continuous_inv⟩
cont... | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | toUnits_homeomorph | If `G` is a group with topological `⁻¹`, then it is homeomorphic to its units. |
Continuous.of_coeHom_comp [Group G] [Monoid H] [TopologicalSpace G] [TopologicalSpace H]
[ContinuousInv G] {f : G →* Hˣ} (hf : Continuous ((Units.coeHom H).comp f)) : Continuous f := by
apply continuous_induced_rng.mpr ?_
refine continuous_prodMk.mpr ⟨hf, ?_⟩
simp_rw [← map_inv]
exact MulOpposite.continuous... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | Continuous.of_coeHom_comp | null |
@[to_additive]
range_embedProduct [Monoid α] :
Set.range (embedProduct α) = {p : α × αᵐᵒᵖ | p.1 * unop p.2 = 1 ∧ unop p.2 * p.1 = 1} :=
Set.range_eq_iff _ _ |>.mpr
⟨fun a ↦ ⟨a.mul_inv, a.inv_mul⟩, fun p hp ↦ ⟨⟨p.1, unop p.2, hp.1, hp.2⟩, rfl⟩⟩
variable [Monoid α] [TopologicalSpace α] [Monoid β] [TopologicalSp... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | range_embedProduct | null |
@[to_additive]
isClosedEmbedding_embedProduct [T1Space α] [ContinuousMul α] :
IsClosedEmbedding (embedProduct α) where
toIsEmbedding := isEmbedding_embedProduct
isClosed_range := by
rw [range_embedProduct]
refine .inter (isClosed_singleton.preimage ?_) (isClosed_singleton.preimage ?_) <;>
fun_prop
@... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | isClosedEmbedding_embedProduct | null |
_root_.Submonoid.units_isCompact [T1Space α] [ContinuousMul α] {S : Submonoid α}
(hS : IsCompact (S : Set α)) : IsCompact (S.units : Set αˣ) := by
have : IsCompact (S ×ˢ S.op) := hS.prod (opHomeomorph.isCompact_preimage.mp hS)
exact isClosedEmbedding_embedProduct.isCompact_preimage this | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | _root_.Submonoid.units_isCompact | null |
@[to_additive prodAddUnits
/-- The topological group isomorphism between the additive units of a product of two
additive monoids, and the product of the additive units of each additive monoid. -/]
_root_.Homeomorph.prodUnits : (α × β)ˣ ≃ₜ αˣ × βˣ where
continuous_toFun :=
(continuous_fst.units_map (MonoidHom.... | def | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | _root_.Homeomorph.prodUnits | The topological group isomorphism between the units of a product of two monoids, and the product
of the units of each monoid. |
@[to_additive]
topologicalGroup_sInf {ts : Set (TopologicalSpace G)}
(h : ∀ t ∈ ts, @IsTopologicalGroup G t _) : @IsTopologicalGroup G (sInf ts) _ :=
letI := sInf ts
{ toContinuousInv :=
@continuousInv_sInf _ _ _ fun t ht => @IsTopologicalGroup.toContinuousInv G t _ <| h t ht
toContinuousMul :=
... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | topologicalGroup_sInf | null |
topologicalGroup_iInf {ts' : ι → TopologicalSpace G}
(h' : ∀ i, @IsTopologicalGroup G (ts' i) _) : @IsTopologicalGroup G (⨅ i, ts' i) _ := by
rw [← sInf_range]
exact topologicalGroup_sInf (Set.forall_mem_range.mpr h')
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | topologicalGroup_iInf | null |
topologicalGroup_inf {t₁ t₂ : TopologicalSpace G} (h₁ : @IsTopologicalGroup G t₁ _)
(h₂ : @IsTopologicalGroup G t₂ _) : @IsTopologicalGroup G (t₁ ⊓ t₂) _ := by
rw [inf_eq_iInf]
refine topologicalGroup_iInf fun b => ?_
cases b <;> assumption | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Pointwise",
"Mathlib.Algebra.Group.Submonoid.Units",
"Mathlib.Algebra.Group.Submonoid.MulOpposite",
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Order.Filter.Bases.Finite",
"Mathlib.Topology.Algebra.Group.Defs",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topolog... | Mathlib/Topology/Algebra/Group/Basic.lean | topologicalGroup_inf | null |
@[ext]
ClosedSubgroup (G : Type u) [Group G] [TopologicalSpace G] extends Subgroup G where
isClosed' : IsClosed carrier | structure | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | ClosedSubgroup | The type of closed subgroups of a topological group. |
@[ext]
ClosedAddSubgroup (G : Type u) [AddGroup G] [TopologicalSpace G] extends
AddSubgroup G where
isClosed' : IsClosed carrier
attribute [to_additive] ClosedSubgroup
attribute [coe] ClosedSubgroup.toSubgroup ClosedAddSubgroup.toAddSubgroup | structure | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | ClosedAddSubgroup | The type of closed subgroups of an additive topological group. |
@[to_additive]
toSubgroup_injective : Function.Injective
(ClosedSubgroup.toSubgroup : ClosedSubgroup G → Subgroup G) :=
fun A B h ↦ by
ext
rw [h]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | toSubgroup_injective | null |
@[to_additive]
instInfClosedSubgroup : Min (ClosedSubgroup G) :=
⟨fun U V ↦ ⟨U ⊓ V, U.isClosed'.inter V.isClosed'⟩⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | instInfClosedSubgroup | null |
instSemilatticeInfClosedSubgroup : SemilatticeInf (ClosedSubgroup G) :=
SetLike.coe_injective.semilatticeInf ((↑) : ClosedSubgroup G → Set G) fun _ _ ↦ rfl
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | instSemilatticeInfClosedSubgroup | null |
normalCore_isClosed (H : Subgroup G) (h : IsClosed (H : Set G)) :
IsClosed (H.normalCore : Set G) := by
rw [normalCore_eq_iInf_conjAct]
push_cast
apply isClosed_iInter
intro g
convert IsClosed.preimage (IsTopologicalGroup.continuous_conj (ConjAct.ofConjAct g⁻¹)) h using 1
exact Set.ext (fun t ↦ Set.mem_... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | normalCore_isClosed | null |
isOpen_of_isClosed_of_finiteIndex (H : Subgroup G) [H.FiniteIndex]
(h : IsClosed (H : Set G)) : IsOpen (H : Set G) := by
apply isClosed_compl_iff.mp
convert isClosed_iUnion_of_finite <| fun (x : {x : (G ⧸ H) // x ≠ QuotientGroup.mk 1})
↦ IsClosed.smul h (Quotient.out x.1)
ext x
refine ⟨fun h ↦ ?_, fun h... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Basic",
"Mathlib.GroupTheory.Index",
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/ClosedSubgroup.lean | isOpen_of_isClosed_of_finiteIndex | null |
@[to_additive
/-- Every topological additive group
in which there exists a compact set with nonempty interior is locally compact. -/]
TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
(K : PositiveCompacts G) : LocallyCompactSpace G :=
let ⟨_x, hx⟩ := K.interior_nonempty
K.isCompact.locallyComp... | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Sets.Compacts"
] | Mathlib/Topology/Algebra/Group/Compact.lean | TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group | Every topological group in which there exists a compact set with nonempty interior
is locally compact. |
@[to_additive]
isInducing_toContinuousMap :
IsInducing (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) := ⟨rfl⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isInducing_toContinuousMap | null |
isEmbedding_toContinuousMap :
IsEmbedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) :=
⟨isInducing_toContinuousMap A B, toContinuousMap_injective⟩
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isEmbedding_toContinuousMap | null |
instContinuousEvalConst : ContinuousEvalConst (ContinuousMonoidHom A B) A B :=
.of_continuous_forget (isInducing_toContinuousMap A B).continuous
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | instContinuousEvalConst | null |
instContinuousEval [LocallyCompactPair A B] :
ContinuousEval (ContinuousMonoidHom A B) A B :=
.of_continuous_forget (isInducing_toContinuousMap A B).continuous
@[to_additive] | instance | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | instContinuousEval | null |
range_toContinuousMap :
Set.range (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) =
{f : C(A, B) | f 1 = 1 ∧ ∀ x y, f (x * y) = f x * f y} := by
refine Set.Subset.antisymm (Set.range_subset_iff.2 fun f ↦ ⟨map_one f, map_mul f⟩) ?_
rintro f ⟨h1, hmul⟩
exact ⟨{ f with map_one' := h1, map_mul' := hmu... | lemma | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | range_toContinuousMap | null |
isClosedEmbedding_toContinuousMap [ContinuousMul B] [T2Space B] :
IsClosedEmbedding (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) where
toIsEmbedding := isEmbedding_toContinuousMap A B
isClosed_range := by
simp only [range_toContinuousMap, Set.setOf_and, Set.setOf_forall]
refine .inter (isClosed... | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isClosedEmbedding_toContinuousMap | null |
@[to_additive]
continuous_of_continuous_uncurry {A : Type*} [TopologicalSpace A]
(f : A → ContinuousMonoidHom B C) (h : Continuous (Function.uncurry fun x y => f x y)) :
Continuous f :=
(isInducing_toContinuousMap _ _).continuous_iff.mpr
(ContinuousMap.continuous_of_continuous_uncurry _ h)
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_of_continuous_uncurry | null |
continuous_comp [LocallyCompactSpace B] :
Continuous fun f : ContinuousMonoidHom A B × ContinuousMonoidHom B C => f.2.comp f.1 :=
(isInducing_toContinuousMap A C).continuous_iff.2 <|
ContinuousMap.continuous_comp'.comp
((isInducing_toContinuousMap A B).prodMap (isInducing_toContinuousMap B C)).continuou... | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_comp | null |
continuous_comp_left (f : ContinuousMonoidHom A B) :
Continuous fun g : ContinuousMonoidHom B C => g.comp f :=
(isInducing_toContinuousMap A C).continuous_iff.2 <|
f.toContinuousMap.continuous_precomp.comp (isInducing_toContinuousMap B C).continuous
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_comp_left | null |
continuous_comp_right (f : ContinuousMonoidHom B C) :
Continuous fun g : ContinuousMonoidHom A B => f.comp g :=
(isInducing_toContinuousMap A C).continuous_iff.2 <|
f.toContinuousMap.continuous_postcomp.comp (isInducing_toContinuousMap A B).continuous
variable (E) in | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | continuous_comp_right | null |
@[to_additive /-- `ContinuousAddMonoidHom _ f` is a functor. -/]
compLeft (f : ContinuousMonoidHom A B) :
ContinuousMonoidHom (ContinuousMonoidHom B E) (ContinuousMonoidHom A E) where
toFun g := g.comp f
map_one' := rfl
map_mul' _g _h := rfl
continuous_toFun := f.continuous_comp_left
variable (A) in | def | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | compLeft | `ContinuousMonoidHom _ f` is a functor. |
@[to_additive /-- `ContinuousAddMonoidHom f _` is a functor. -/]
compRight {B : Type*} [CommGroup B] [TopologicalSpace B] [IsTopologicalGroup B]
(f : ContinuousMonoidHom B E) :
ContinuousMonoidHom (ContinuousMonoidHom A B) (ContinuousMonoidHom A E) where
toFun g := f.comp g
map_one' := ext fun _a => map_one... | def | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | compRight | `ContinuousMonoidHom f _` is a functor. |
@[to_additive]
isClosedEmbedding_coe : IsClosedEmbedding ((⇑) : (A →ₜ* B) → A → B) :=
ContinuousMap.isHomeomorph_coe.isClosedEmbedding.comp <| isClosedEmbedding_toContinuousMap ..
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | isClosedEmbedding_coe | null |
@[to_additive]
locallyCompactSpace_of_equicontinuousAt (U : Set X) (V : Set Y)
(hU : IsCompact U) (hV : V ∈ nhds (1 : Y))
(h : EquicontinuousAt (fun f : {f : X →* Y | Set.MapsTo f U V} ↦ (f : X → Y)) 1) :
LocallyCompactSpace (ContinuousMonoidHom X Y) := by
replace h := equicontinuous_of_equicontinuousAt_o... | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | locallyCompactSpace_of_equicontinuousAt | null |
locallyCompactSpace_of_hasBasis (V : ℕ → Set Y)
(hV : ∀ {n x}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1))
(hVo : Filter.HasBasis (nhds 1) (fun _ ↦ True) V) :
LocallyCompactSpace (ContinuousMonoidHom X Y) := by
obtain ⟨U0, hU0c, hU0o⟩ := exists_compact_mem_nhds (1 : X)
let U_aux : ℕ → {S : Set X | S ∈ nhds 1... | theorem | Topology | [
"Mathlib.Topology.Algebra.ContinuousMonoidHom",
"Mathlib.Topology.Algebra.Equicontinuity",
"Mathlib.Topology.Algebra.Group.Compact",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Topology.UniformSpace.Ascoli"
] | Mathlib/Topology/Algebra/Group/CompactOpen.lean | locallyCompactSpace_of_hasBasis | null |
ContinuousNeg (G : Type u) [TopologicalSpace G] [Neg G] : Prop where
continuous_neg : Continuous fun a : G => -a
attribute [continuity, fun_prop] ContinuousNeg.continuous_neg | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousNeg | Basic hypothesis to talk about a topological additive group. A topological additive group
over `M`, for example, is obtained by requiring the instances `AddGroup M` and
`ContinuousAdd M` and `ContinuousNeg M`. |
@[to_additive (attr := continuity)]
ContinuousInv (G : Type u) [TopologicalSpace G] [Inv G] : Prop where
continuous_inv : Continuous fun a : G => a⁻¹
attribute [continuity, fun_prop] ContinuousInv.continuous_inv
export ContinuousInv (continuous_inv)
export ContinuousNeg (continuous_neg) | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousInv | Basic hypothesis to talk about a topological group. A topological group over `M`, for example,
is obtained by requiring the instances `Group M` and `ContinuousMul M` and
`ContinuousInv M`. |
@[to_additive
/-- If a function converges to a value in an additive topological group, then its
negation converges to the negation of this value. -/]
Filter.Tendsto.inv {f : α → G} {l : Filter α} {y : G} (h : Tendsto f l (𝓝 y)) :
Tendsto (fun x => (f x)⁻¹) l (𝓝 y⁻¹) :=
(continuous_inv.tendsto y).comp h
vari... | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Filter.Tendsto.inv | If a function converges to a value in a multiplicative topological group, then its inverse
converges to the inverse of this value.
For the version in topological groups with zero (including topological fields)
assuming additionally that the limit is nonzero, use `Filter.Tendsto.inv₀`. |
Continuous.inv (hf : Continuous f) : Continuous fun x => (f x)⁻¹ :=
continuous_inv.comp hf
@[to_additive]
nonrec theorem ContinuousWithinAt.inv (hf : ContinuousWithinAt f s x) :
ContinuousWithinAt (fun x => (f x)⁻¹) s x :=
hf.inv
@[to_additive (attr := fun_prop)]
nonrec theorem ContinuousAt.inv (hf : Continuous... | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Continuous.inv | null |
ContinuousOn.inv (hf : ContinuousOn f s) : ContinuousOn (fun x => (f x)⁻¹) s := fun x hx ↦
(hf x hx).inv | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousOn.inv | null |
IsTopologicalAddGroup (G : Type u) [TopologicalSpace G] [AddGroup G] : Prop
extends ContinuousAdd G, ContinuousNeg G | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | IsTopologicalAddGroup | A topological (additive) group is a group in which the addition and negation operations are
continuous.
When you declare an instance that does not already have a `UniformSpace` instance,
you should also provide an instance of `UniformSpace` and `IsUniformAddGroup` using
`IsTopologicalAddGroup.toUniformSpace` and `isUn... |
@[to_additive]
IsTopologicalGroup (G : Type*) [TopologicalSpace G] [Group G] : Prop
extends ContinuousMul G, ContinuousInv G | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | IsTopologicalGroup | A topological group is a group in which the multiplication and inversion operations are
continuous.
When you declare an instance that does not already have a `UniformSpace` instance,
you should also provide an instance of `UniformSpace` and `IsUniformGroup` using
`IsTopologicalGroup.toUniformSpace` and `isUniformGroup... |
ContinuousSub (G : Type*) [TopologicalSpace G] [Sub G] : Prop where
continuous_sub : Continuous fun p : G × G => p.1 - p.2 | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousSub | A typeclass saying that `p : G × G ↦ p.1 - p.2` is a continuous function. This property
automatically holds for topological additive groups but it also holds, e.g., for `ℝ≥0`. |
@[to_additive existing]
ContinuousDiv (G : Type*) [TopologicalSpace G] [Div G] : Prop where
continuous_div' : Continuous fun p : G × G => p.1 / p.2
@[to_additive] | class | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousDiv | A typeclass saying that `p : G × G ↦ p.1 / p.2` is a continuous function. This property
automatically holds for topological groups. Lemmas using this class have primes.
The unprimed version is for `GroupWithZero`. |
@[to_additive sub]
Filter.Tendsto.div' {f g : α → G} {l : Filter α} {a b : G} (hf : Tendsto f l (𝓝 a))
(hg : Tendsto g l (𝓝 b)) : Tendsto (fun x => f x / g x) l (𝓝 (a / b)) :=
(continuous_div'.tendsto (a, b)).comp (hf.prodMk_nhds hg)
variable {f g : X → G} {s : Set X} {x : X}
@[to_additive (attr := fun_prop) s... | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Filter.Tendsto.div' | null |
ContinuousWithinAt.div' (hf : ContinuousWithinAt f s x) (hg : ContinuousWithinAt g s x) :
ContinuousWithinAt (fun x => f x / g x) s x :=
Filter.Tendsto.div' hf hg
@[to_additive (attr := fun_prop) sub] | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousWithinAt.div' | null |
ContinuousOn.div' (hf : ContinuousOn f s) (hg : ContinuousOn g s) :
ContinuousOn (fun x => f x / g x) s := fun x hx => (hf x hx).div' (hg x hx)
@[to_additive (attr := continuity, fun_prop) sub] | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | ContinuousOn.div' | null |
Continuous.div' (hf : Continuous f) (hg : Continuous g) : Continuous fun x => f x / g x :=
continuous_div'.comp₂ hf hg | theorem | Topology | [
"Mathlib.Topology.Algebra.Monoid.Defs"
] | Mathlib/Topology/Algebra/Group/Defs.lean | Continuous.div' | null |
GroupTopology (α : Type u) [Group α] : Type u
extends TopologicalSpace α, IsTopologicalGroup α | structure | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | GroupTopology | A group topology on a group `α` is a topology for which multiplication and inversion
are continuous. |
AddGroupTopology (α : Type u) [AddGroup α] : Type u
extends TopologicalSpace α, IsTopologicalAddGroup α
attribute [to_additive] GroupTopology | structure | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | AddGroupTopology | An additive group topology on an additive group `α` is a topology for which addition and
negation are continuous. |
@[to_additive /-- A version of the global `continuous_add` suitable for dot notation. -/]
continuous_mul' (g : GroupTopology α) :
haveI := g.toTopologicalSpace
Continuous fun p : α × α => p.1 * p.2 := by
letI := g.toTopologicalSpace
haveI := g.toIsTopologicalGroup
exact continuous_mul | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | continuous_mul' | A version of the global `continuous_mul` suitable for dot notation. |
@[to_additive /-- A version of the global `continuous_neg` suitable for dot notation. -/]
continuous_inv' (g : GroupTopology α) :
haveI := g.toTopologicalSpace
Continuous (Inv.inv : α → α) := by
letI := g.toTopologicalSpace
haveI := g.toIsTopologicalGroup
exact continuous_inv
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | continuous_inv' | A version of the global `continuous_inv` suitable for dot notation. |
toTopologicalSpace_injective :
Function.Injective (toTopologicalSpace : GroupTopology α → TopologicalSpace α) :=
fun f g h => by
cases f
cases g
congr
@[to_additive (attr := ext)] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | toTopologicalSpace_injective | null |
ext' {f g : GroupTopology α} (h : f.IsOpen = g.IsOpen) : f = g :=
toTopologicalSpace_injective <| TopologicalSpace.ext h | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | ext' | null |
@[to_additive
/-- Given `f : α → β` and a topology on `α`, the coinduced additive group topology on `β`
is the finest topology such that `f` is continuous and `β` is a topological additive group. -/]
coinduced {α β : Type*} [t : TopologicalSpace α] [Group β] (f : α → β) : GroupTopology β :=
sInf { b : GroupTopolo... | def | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | coinduced | The ordering on group topologies on the group `γ`. `t ≤ s` if every set open in `s` is also open
in `t` (`t` is finer than `s`). -/
@[to_additive
/-- The ordering on group topologies on the group `γ`. `t ≤ s` if every set open in `s`
is also open in `t` (`t` is finer than `s`). -/]
instance : PartialOrder (GroupTop... |
coinduced_continuous {α β : Type*} [t : TopologicalSpace α] [Group β] (f : α → β) :
Continuous[t, (coinduced f).toTopologicalSpace] f := by
rw [continuous_sInf_rng]
rintro _ ⟨t', ht', rfl⟩
exact continuous_iff_coinduced_le.2 ht' | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/GroupTopology.lean | coinduced_continuous | null |
@[to_additive /-- Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point. -/]
smul_singleton_mem_nhds_of_sigmaCompact
{U : Set G} (hU : U ∈ 𝓝 1) (x : X) : ... | theorem | Topology | [
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/OpenMapping.lean | smul_singleton_mem_nhds_of_sigmaCompact | Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. |
@[to_additive /-- Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open. This is a version of the open mapping theorem, valid
notably for the action of a sigma-compact locally compact group on a locally compact space. -/]
isOpenMap_smul_of_sigmaCompact... | theorem | Topology | [
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/OpenMapping.lean | isOpenMap_smul_of_sigmaCompact | Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open. This is a version of the open mapping theorem, valid notably for the
action of a sigma-compact locally compact group on a locally compact space. |
@[to_additive]
MonoidHom.isOpenMap_of_sigmaCompact
{H : Type*} [Group H] [TopologicalSpace H] [BaireSpace H] [T2Space H] [ContinuousMul H]
(f : G →* H) (hf : Function.Surjective f) (h'f : Continuous f) :
IsOpenMap f := by
let A : MulAction G H := MulAction.compHom _ f
have : ContinuousSMul G H := contin... | theorem | Topology | [
"Mathlib.Topology.Baire.Lemmas",
"Mathlib.Topology.Algebra.Group.Pointwise"
] | Mathlib/Topology/Algebra/Group/OpenMapping.lean | MonoidHom.isOpenMap_of_sigmaCompact | A surjective morphism of topological groups is open when the source group is sigma-compact and
the target group is a Baire space (for instance a locally compact group). |
@[to_additive]
subset_interior_smul : interior s • interior t ⊆ interior (s • t) :=
(Set.smul_subset_smul_right interior_subset).trans subset_interior_smul_right | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_smul | null |
@[to_additive]
IsClosed.smul_left_of_isCompact (ht : IsClosed t) (hs : IsCompact s) :
IsClosed (s • t) := by
have : ∀ x ∈ s • t, ∃ g ∈ s, g⁻¹ • x ∈ t := by
rintro x ⟨g, hgs, y, hyt, rfl⟩
refine ⟨g, hgs, ?_⟩
rwa [inv_smul_smul]
choose! f hf using this
refine isClosed_of_closure_subset (fun x hx ↦ ?... | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsClosed.smul_left_of_isCompact | null |
MulAction.isClosedMap_quotient [CompactSpace α] :
letI := orbitRel α β
IsClosedMap (Quotient.mk' : β → Quotient (orbitRel α β)) := by
intro t ht
rw [← isQuotientMap_quotient_mk'.isClosed_preimage,
MulAction.quotient_preimage_image_eq_union_mul]
convert ht.smul_left_of_isCompact (isCompact_univ (X := α... | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | MulAction.isClosedMap_quotient | null |
@[to_additive]
IsOpen.mul_left : IsOpen t → IsOpen (s * t) :=
IsOpen.smul_left
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | IsOpen.mul_left | null |
subset_interior_mul_right : s * interior t ⊆ interior (s * t) :=
subset_interior_smul_right
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_mul_right | null |
subset_interior_mul : interior s * interior t ⊆ interior (s * t) :=
subset_interior_smul
@[to_additive] | theorem | Topology | [
"Mathlib.Topology.Algebra.Group.Basic"
] | Mathlib/Topology/Algebra/Group/Pointwise.lean | subset_interior_mul | null |
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